设a,b,c均为非零实数,且ab=2(a+b),bc=3(b+c),ca=4(c+a),试求(a-1)(b-1)(c-1)

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设a,b,c均为非零实数,且ab=2(a+b),bc=3(b+c),ca=4(c+a),试求(a-1)(b-1)(c-1)

设a,b,c均为非零实数,且ab=2(a+b),bc=3(b+c),ca=4(c+a),试求(a-1)(b-1)(c-1)
设a,b,c均为非零实数,且ab=2(a+b),bc=3(b+c),ca=4(c+a),试求(a-1)(b-1)(c-1)

设a,b,c均为非零实数,且ab=2(a+b),bc=3(b+c),ca=4(c+a),试求(a-1)(b-1)(c-1)
ab=2(a+b),bc=3(b+c),ca=4(c+a),
(a+b)/(ab)=1/2,(b+c)/(bc)=1/3,(c+a)/(ca)=1/4
1/a+1/b=1/2
1/b+1/c=1/3
1/c+1/a=1/4
3式相加得:
1/a+1/b+1/c=13/24
1/a=5/24,a=24/5
1/b=7/24,b=24/7
1/c=1/24,c=24
(a-1)(b-1)(c-1)
=(19/5)(17/7)(23)
=7429/35

由已知变形得:
(a+b)/ab=1/2
(b+c)/bc=1/3
(c+a)/ca=1/4
进一步变形为:
1/a+1/b=1/2········①
1/b+1/c=1/3········②
1/c+1/a=1/4········③
①+②+③,得:
2(1/a+1/b+1/c)=1/2+1/3+1/4=1...

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由已知变形得:
(a+b)/ab=1/2
(b+c)/bc=1/3
(c+a)/ca=1/4
进一步变形为:
1/a+1/b=1/2········①
1/b+1/c=1/3········②
1/c+1/a=1/4········③
①+②+③,得:
2(1/a+1/b+1/c)=1/2+1/3+1/4=13/12
1/a+1/b+1/c=13/24
上式分别减去①、②、③,得:
1/c=1/24
得:c=24,
1/a=5/24
得:a=24/5,
1/b=7/24
得:b=24/7;
所以
a+b+c
=24/5+24/7+24
=1128/35

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由已知变形得:
(a+b)/ab=1/2
(b+c)/bc=1/3
(c+a)/ca=1/4
进一步变形为:
(1/a)+(1/b)=1/2········①
(1/b)+(1/c)=1/3········②
(1/c)+(1/a)=1/4········③
①+②+③,得:
2(1/a+1/b+1/c)=1...

全部展开

由已知变形得:
(a+b)/ab=1/2
(b+c)/bc=1/3
(c+a)/ca=1/4
进一步变形为:
(1/a)+(1/b)=1/2········①
(1/b)+(1/c)=1/3········②
(1/c)+(1/a)=1/4········③
①+②+③,得:
2(1/a+1/b+1/c)=1/2+1/3+1/4=13/12
1/a+1/b+1/c=13/24
上式分别减去①、②、③,得:
1/c=1/24
得:c=24,
1/a=5/24
得:a=24/5,
1/b=7/24
得:b=24/7;
所以
(a-1)(b-1)(c-1)
=19/5 * 17/7 *23
=7429/35

收起